Weak Laplace principles on topological spaces.

We address the missing analog of vague convergence in the weak-converge large-deviations analogy. Specifically, we introduce the weak Laplace principle and show it implies both the well-known weak LDP and the Laplace principle lower bound. Both the weak LDP and weak Laplace principle hold in settings where there is no exponential tightness and imply the LDP when exponential tightness holds as well as the Laplace principle when, in addition, the space is Polish. As a side effect, we also generalize Bryc’s lemma. Whereas vague convergence is only defined on locally compact spaces, our definition of the weak Laplace principle holds on general topological spaces.